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arXiv:1712.04021 · Physical review. B./Physical review. B

Study of anyon condensation and topological phase transitions from a Z4 topological phase using the projected entangled pair states approach

Mohsin Iqbal, Kasper Duivenvoorden, Norbert Schuch

Published May 14, 2018• 37 citations
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Abstract

We use projected entangled pair states (PEPS) to study topological quantum phase transitions. The local description of topological order in the PEPS formalism allows us to set up order parameters which measure condensation and deconfinement of anyons and serve as substitutes for conventional order parameters. We apply these order parameters, together with anyon-anyon correlation functions and some further probes, to characterize topological phases and phase transitions within a family of models based on a ${\mathbb{Z}}_{4}$ symmetry, which contains ${\mathbb{Z}}_{4}$ quantum double, toric code, double semion, and trivial phases. We find a diverse phase diagram which exhibits a variety of different phase transitions of both first and second order which we comprehensively characterize, including direct transitions between the toric code and the double semion phase.

DOI
10.1103/physrevb.97.195124

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We use projected entangled pair states (PEPS) to study topological quantum phase transitions. The local description of topological order in the PEPS formalism allows us to set up order parameters which measure condensation and deconfinement of anyons and serve as substitutes for conventional order parameters. We apply these order parameters, together with anyon-anyon correlation functions and some further probes, to characterize topological phases and phase transitions within a family of models based on a Z4{\mathbb{Z}}_{4}Z4​ symmetry, which contains Z4{\mathbb{Z}}_{4}Z4​ quantum double, toric code, double semion, and trivial phases. We find a diverse phase diagram which exhibits a variety of different phase transitions of both first and second order which we comprehensively characterize, including direct transitions between the toric code and the double semion phase.
Topological phases exhibit unconventional order that cannot be detected by any local order parameter. In the framework of Projected Entangled Pair States(PEPS), topological order is characterized by an entanglement symmetry of the local tensor which describes the model. This symmetry can take the form of a tensor product of group representations, or in the more general case a correlated symmetry action in the form of a Matrix Product Operator(MPO), which encompasses all string-net models. Among other things, these entanglement symmetries allow for the description of ground states and anyon excitations. Recently, the idea has been put forward to use those symmetries and the anyonic objects they describe as order parameters for probing topological phase transitions, and the applicability of this idea has been demonstrated for Abelian groups. In this paper, we extend this construction to the domain of non-Abelian models with MPO symmetries, and use it to study the breakdown of topological order in the double Fibonacci (DFib) string-net and its Galois conjugate, the non-hermitian double Yang-Lee (DYL) string-net. We start by showing how to construct topological order parameters for condensation and deconfinement of anyons using the MPO symmetries. Subsequently, we set up interpolations from the DFib and the DYL model to the trivial phase, and show that these can be mapped to certain restricted solid on solid(RSOS) models, which are equivalent to the ((5±5)/2)((5\pm\sqrt{5})/2)((5±5​)/2)-state Potts model, respectively. The known exact solutions of the statistical models allow us to locate the critical points, and to predict the critical exponents for the order parameters. We complement this by numerical study of the phase transitions, which fully confirms our theoretical predictions; remarkably, we find that both models exhibit a duality between the order parameters for condensation and deconfinement.
Topological phases exhibit unconventional order that cannot be detected by any local order parameter. In the framework of projected entangled pair states, topological order is characterized by an entanglement symmetry of the local tensor that describes the model. This symmetry can take the form of a tensor product of group representations [for quantum double models D(G)D(G)D(G) of a group G]G]G], or in the more general case a correlated symmetry action in the form of a matrix product operator (MPO), which encompasses all string-net models, including those that are not quantum double models. Among other things, these entanglement symmetries allow for the succinct description of ground states and topological excitations (anyons). Recently, the idea has been put forward to use those symmetries and the anyonic objects they describe as order parameters for probing topological phase transitions, and the applicability of this idea has been demonstrated for Abelian groups. In this paper, we extend this construction to the domain of non-Abelian models with MPO symmetries, and we use it to study the breakdown of topological order in the double Fibonacci (DFib) string-net model and its Galois conjugate, namely the non-Hermitian double Yang-Lee (DYL) string-net model. We start by showing how to construct topological order parameters for condensation and deconfinement of anyons using the MPO symmetries. Subsequently, we set up interpolations from the DFib and the DYL model to the trivial phase, and we show that these can be mapped to certain restricted solid on solid (RSOS) models, which are equivalent to the $[(5\ifmmode\pm\else\textpm\fi{}\sqrt{5})/2]$-state Potts model, respectively. Moreover, the order parameter for condensation maps to the RSOS order parameter. The known exact solutions of the statistical models subsequently allow us to locate the critical points of the models, and to predict the critical exponents for the order parameters from conformal field theory. We complement this by numerical study of the phase transitions, which fully confirms our theoretical predictions; remarkably, we find that both models exhibit a duality between the behavior of order parameters for condensation and deconfinement.
Symmetrization of topologically ordered wave functions is a powerful method for constructing new topological models. Here we study wave functions obtained by symmetrizing quantum double models of a group GGG in the projected entangled pair states (PEPS) formalism. We show that symmetrization naturally gives rise to a larger symmetry group $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{G}$ which is always non-Abelian. We prove that by symmetrizing on sufficiently large blocks, one can always construct wave functions in the same phase as the double model of $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{G}$. In order to understand the effect of symmetrization on smaller patches, we carry out numerical studies for the toric code model, where we find strong evidence that symmetrizing on individual spins gives rise to a critical model which is at the phase transitions of two inequivalent toric codes, obtained by anyon condensation from the double model of $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{G}$.
Topological phase transition of deformed Z3{\mathbb Z}_3Z3​ toric code
We investigate topological phase transitions in a family of deformed Z3\mathbb Z_3Z3​ toric-code wavefunctions prepared from a cluster state by local deformations and projective measurements. Their norms map to the Q=3Q=3Q=3 Potts model for single-parameter deformations and to a three-state Ashkin--Teller-like (AT3_33​) construction with two independent four-spin couplings in the general case. Projected entangled-pair-state (PEPS) and variational uniform matrix-product-state (VUMPS) calculations identify the toric-code (TC) phase and phases in which electric (eee) anyons are confined or condensed. These phases are separated by critical structures with central charges c=4/5c=4/5c=4/5, 8/58/58/5, and isolated c=1c=1c=1 antiferromagnetic (AFM) endpoints. A normalized finite-distance eee-anyon pair-state norm provides a Fredenhagen--Marcu-type check of the confinement boundary, while the topological data of the quantum double D(Z3)D(\mathbb Z_3)D(Z3​) imply a topological entanglement entropy γ=log⁡3γ=\log3γ=log3 throughout the gapped toric-code phase. Relative to the Z2\mathbb Z_2Z2​ case, the absence of sign-change folding leaves the AFM endpoints unfolded, and the extreme deformation reaches square ice with an emergent U(1)U(1)U(1) one-form symmetry, Hilbert-space fragmentation, and exact scar configurations.
The focus of our investigations in this thesis is quantum spin systems in two dimensions. We examine phase transitions between topologically distinct phases of quantum matter using the framework of tensor network states. We study the phenomena of anyon condensation and confinement in the context of tensor network states where these two notions offer a robust probe to characterize the universal features of quantum phase transitions. We do a comprehensive study of the numerical methods enabled by the tensor network formalism for the study of quantum phase transitions. We map out the phase diagram of certain exotic phases of quantum matter (namely the D(mathbbZ4)D(\\mathbb{Z}_4)D(mathbbZ4​) quantum double, the toric code, and the double semion model) and identify the order and the universality classes of the phase transitions between these distinct topological phases. We analyze the phase boundaries of the toric code and the double semion model which exhibit first and second order phase transitions. We investigate the class of mathbbZ2\\mathbb{Z}_2mathbbZ2​-invariant tensor network states while taking in to account the spin rotation and lattice symmetries. The resulting tensor network states allow us to map the phase diagram of mathbbZ2\\mathbb{Z}_2mathbbZ2​ spin liquids. The behavior of the system at the phase boundaries of mathbbZ2\\mathbb{Z}_2mathbbZ2​ spin liquid is governed by the condensation of spinons and visions. We also present our preliminary findings regarding the mathbbZ4\\mathbb{Z}_4mathbbZ4​-invariant tensors with SU(2)SU(2)SU(2) symmetry. This approach enables us to study the spin liquid states of the toric code and the double semion model in a unified framework. We give a local tensor description for the approximate ground states of the Heisenberg antiferromagnet on the kagome lattice by using the variational manifolds of just three and five parameters in the gapped mathbbZ2\\mathbb{Z}_2mathbbZ2​ spin liquid phase. The approximation of the ground states we construct have an energy density that is remarkably close to the results from the state-of-the-art density matrix renormalization group and the exact diagonalization method. By analyzing the deconfinement of anyonic excitations, we also present our findings regarding the vicinity of the variational ground state in gapped mathbbZ2\\mathbb{Z}_2mathbbZ2​ spin liquid phase.
Topological phase transitions beyond anyon condensation remain poorly understood. A notable example is the transition between the toric code (TC) and double semion (DS) phases, which has two distinct Z 2 topological orders in (2+1) dimensions [ ( 2 + 1 ) D ]. Previous studies reveal that the transition between them can be either first order or via an intermediate phase, thus the existence of a direct continuous transition between them remains a long-standing problem. Motivated by the fact that both phases can arise from condensing distinct anyons in the Z 4 topological order, we introduce a perturbed Z 4 quantum double (QD) model to study the TC-DS transition. We confirm the existence of a continuous ( 2 + 1 ) D XY* transition between the TC and DS phases by mapping it to a two-coupled quantum Ising model. Importantly, using the condensation order parameters and the area law coefficients of the Wilson loops, we further reveal that Z 4 anyons, fractionalized from the Z 2 topological orders, become deconfined at the transition between Z 2 topologically ordered phases. Our results open a path toward developing a theoretical framework for topological phase transitions beyond anyon condensation.